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Final Review: Core Question Templates Final 复习：Core Questions 秒题模板

How To Use This Page 怎么用这页复习

This page is for Seen Questions Core Group Q1-Q25. Each template below uses the real question shapes from the first 25 core questions, then gives the recognition rule, the working steps, and the answer format. By question number, odd questions are free and even questions are notes.

这一页专门对应 Seen Questions Core Group 前 25 道题。每个模板都放真实题型、识别方法、做题步骤和答案格式。细讲部分按题号“一题免费，一题会员”开放：奇数题免费，偶数题会员。目标是：看到题能马上知道用哪个套路。

Exam rule: recognise the shape -> choose the test/formula -> write the first line. 考试规则：先认题型 -> 选公式/判别法 -> 立刻写第一行。

Calculus Mantras To Memorise First 微积分秒题模板：先背这个

Mixed Partial: Kill Irrelevant Terms混合偏导秒杀

If the derivative includes z, first remove every term with no z.如果题目有对 z 求偏导，先看哪些项含 z。不含 z 的项对 z 求导后全部变 0。

d/dz(6*x*y*z) = 6*x*y -> d/dy = 6*x -> d/dx = 6

Chain Rule: Outside Then Inside链式法则外壳法

For F(x,y)=f(u), outside f becomes f', inside u stays copied, then multiply by derivative of u.遇到 F(x,y)=f(u)，外面 f 变 f'，里面 u 原样抄，再乘 u 对目标变量的导数。

u=x^2+x*y+5*y^2
F_y=(x+10*y)f'(u), F_x=(2*x+y)f'(u)

Recover f(t) From F(x,x)从 F(x,x) 找 f(t)

Put y=x, simplify the inside, then rename the inside as t.看到 F(x,x)，先把 y 全换成 x。里面变成什么，就令它等于 t。

F(x,x)=f(7*x^2)=-4*x^2
t=7*x^2 -> f(t)=-4*t/7

PDF Normalisation概率密度归一化

A pdf must have total area 1. Expected value adds one extra x.pdf 的总面积必须等于 1。求期望就是把 x 乘进 pdf 再积分。

find alpha: integral f(x) dx = 1
E(X)=integral x*f(x) dx

Continuous Probability连续型概率

Single points have probability 0. Symmetry about x=5 means equal area on both sides.连续型随机变量：单点概率永远是 0。关于 x=5 对称时，左边面积等于右边面积。

P(X=4)=P(X=5)=0
P(X<=5)=P(X>=5)

Series With log(k)带 log(k) 的级数

For 1/(k^p log k), look at p first. The log is weak.看到 1/(k^p log k)，先看 p。log 很弱，p<1 时救不了，直接发散。

p<1 diverges; p>1 converges;
p=1 then inspect log power

Linear Algebra Mantras To Memorise First 线代秒题模板：先背这个

Question Shape题型长相 What To Memorise背什么 First Line第一步写什么
Linear transformation with e1,e2,e3线性变换给 e1,e2,e3 T can split sums; T(0)=0.T 可以拆加法；T(0)=0。 (0,1,1)=e2+e3
Matrix from graph从图读变换矩阵 Columns of P are T(e1), T(e2).矩阵 P 的两列就是 T(e1), T(e2)。 e1 = alpha*A + beta*B
Span / not span张成 / 不张成 Span: every target can be made. Not span: one target cannot be made.张成：每个目标都能拼。不张成：存在一个目标拼不出。 for every x, there exist coefficients
Independent / dependent线性无关 / 线性相关 Independent: only all-zero coefficients make 0. Dependent: non-zero coefficients can make 0.无关：凑 0 只能全 0。相关：不全 0 也能凑 0。 c1*v1+c2*v2+...=0
Basis in R^nR^n 的 basis A basis has exactly n vectors, spans, and is independent.basis 必须刚好 n 个，还要 span 和 independent。 R^5 needs 5 vectors
Polynomial vectors and pivots多项式向量和 pivot Use coefficient vectors; pivot columns give basis columns.把多项式看成系数向量；pivot columns 对应 basis。 p=a+b*x+c*x^2 -> <a,b,c>
Diagonalisation A=MDM^-1对角化 A=MDM^-1 Diagonal entry i of D must match column i of M.D 第几个 eigenvalue，M 第几列就要放对应 eigenvector。 D_ii lambda -> M column i eigenvector
M invertible checkM 可逆检查 Columns of M cannot repeat the same direction.M 的列不能重复方向；同方向就是线性相关，M 不可逆。 col2 = 2*col1 -> reject
Not diagonalisable不可对角化 Repeated eigenvalue needs enough independent eigenvectors.重复特征值时，要检查独立特征向量够不够。 nullity(C-lambda*I)

Final Memory Sheet 最后背诵表：考前看这个

  • Converges means the infinite process has a finite answer.converges = 最后有有限答案。
  • Ratio test: L<1 converges, L>1 diverges, L=1 no conclusion.ratio test：L<1 收敛，L>1 发散，L=1 没结论。
  • Alternating test: alternating, decreasing, limit 0.alternating test：正负交替、越来越小、极限为 0。
  • Continuous probability: intervals have area; single points have probability 0.连续概率：区间才有面积，单点概率为 0。
  • Conditional probability: restrict to the given world, then divide by that world's probability.条件概率：先缩小到 given 的世界，最后除以这个世界的概率。
  • Normal table: table entries are left areas. Conditional normal uses middle area divided by the given tail.正态表：表里是左边面积。条件正态题用中间面积除以 given 的尾巴面积。
  • Tangent plane: z=z0+Hx(x-x0)+Hy(y-y0).切平面必背：z=z0+Hx(x-x0)+Hy(y-y0)。
  • Linear approximation: Delta F is approximately Fx Delta x plus Fy Delta y.线性近似：ΔF≈FxΔx+FyΔy。
  • Exact ODE: M+N y'=0 means M=Hx and N=Hy.exact ODE：M+N y'=0 就读成 M=Hx, N=Hy。
  • Diagonalisation: D order and M column order must match.对角化：D 的顺序和 M 的列顺序必须一一对应。
  • M must be invertible, so its columns cannot be repeated directions.M 必须可逆，所以列向量不能重复方向。
  • Maple syntax: use * for multiplication and <a,b> for vectors.Maple 写法：乘法必须写 *，向量用 <a,b>。

Detailed Calculus Walkthroughs 下面开始细讲：微积分题型

Q1 Real Type Free

Mixed Partial Derivative With One z Term真实题 1：三阶混合偏导，只有一项含 z

Real shape: f(x,y,z)=sinh(y)/sqrt(y+x)+6xyz+arccos(x/sqrt(y^2+x^2)). Calculate partial^3 f / partial x partial y partial z.

真实题型：f(x,y,z)=sinh(y)/sqrt(y+x)+6xyz+arccos(x/sqrt(y^2+x^2))，求 ∂³f/∂x∂y∂z。

  1. First look for z. Terms without z die after the z-derivative.第一眼找 z：不含 z 的项，对 z 求导后全部变 0。
  2. Only 6xyz survives.只剩 6xyz。
  3. Differentiate by z, then y, then x: 6xyz -> 6xy -> 6x -> 6.按 z、y、x 求：6xyz -> 6xy -> 6x -> 6。
Maple answer: 6
记忆：看到 ∂/∂z，先杀掉所有不含 z 的项。Memory: if the derivative includes z, eliminate every term with no z first.
Q7 Real Type Free

Chain Rule: F(x,y)=f(inner)真实题 7：链式法则外面求导 × 里面求导

Real shape: F(x,y)=f(x^2+x*y+5*y^2). Find g(x,y), h(x,y), and then recover f(t) if F(x,x)=-4x^2.

真实题型：F(x,y)=f(x^2+x*y+5*y^2)。先找 g(x,y) 和 h(x,y)，再用 F(x,x)=-4x^2 找 f(t)。

  1. Set u=x^2+x*y+5*y^2.设里面 u=x^2+x*y+5*y^2。
  2. F_y=f'(u)*u_y=(x+10y)f'(u), so g=x+10y.F_y=f'(u)*u_y=(x+10y)f'(u)，所以 g=x+10y。
  3. F_x=f'(u)*u_x=(2x+y)f'(u), so h=2x+y.F_x=f'(u)*u_x=(2x+y)f'(u)，所以 h=2x+y。
  4. Put y=x: F(x,x)=f(7x^2)=-4x^2. Let t=7x^2, so f(t)=-4t/7.把 y=x 代进去：F(x,x)=f(7x^2)=-4x^2。令 t=7x^2，得到 f(t)=-4t/7。
g = x+10*y
h = 2*x+y
f(t) = -4*t/7
F(x,y) = -4*(x^2+x*y+5*y^2)/7
记忆：外面 f 变 f'，里面原样保留，再乘里面对目标变量的导数。Memory: outside becomes f', inside stays unchanged, then multiply by derivative of inside.
Q8 Real Type Notes

Continuous PDF: Find alpha and E(X)真实题 8：连续型 pdf 求 alpha 和期望

Real shape: f(x)=alpha*(x^2+3x-4) for -4<=x<=1, and 0 otherwise.

真实题型：f(x)=alpha*(x^2+3x-4)，范围是 -4<=x<=1，其他地方为 0。

  1. PDF total area must be 1: alpha * integral_{-4}^{1}(x^2+3x-4) dx = 1.pdf 总面积必须等于 1：alpha * ∫_{-4}^{1}(x^2+3x-4) dx = 1。
  2. The integral is -125/6, so alpha=-6/125.积分结果是 -125/6，所以 alpha=-6/125。
  3. Expected value: E(X)=integral x*f(x) dx.期望公式：E(X)=∫x*f(x) dx。
alpha = -6/125
E(X) = -3/2
记忆：求 alpha 是“总面积=1”；求期望是“多乘一个 x”。Memory: alpha uses total area 1; expectation adds one extra x.
Q10 Real Type Notes

Continuous Symmetric Distribution真实题 10：连续型 + 关于 x=5 对称

Real shape: f(x)=g(x) on 3<=x<=7, where g is symmetric about x=5. Select statements that must be true.

真实题型：f(x)=g(x) 在 3<=x<=7 上，且 g 关于 x=5 对称。判断哪些概率语句一定对。

  1. Continuous means P(X=a)=0 for any exact point.连续型：任何单点概率 P(X=a)=0。
  2. Symmetric about 5 means left area equals right area.关于 5 对称：左边面积等于右边面积。
  3. Support is [3,7], so intervals outside 7 add no extra probability.范围是 [3,7]，超过 7 的区间没有额外概率。
True statements: P(X=5) >= P(X=4), and P(X<=5)=P(X>=5)
记忆：连续型看面积，点没有宽度，所以 P(X=4)=P(X=5)=0。Memory: continuous probability uses area; a single point has zero width.
Q11 Real Type Free

Series With k^(1/3) log(k)真实题 11：带 log(k) 的级数发散判断

Real shape: sum_{k=2}^{infinity} 1/(k^(1/3)*log(k)). Decide converge/diverge and select usable tests.

真实题型：sum_{k=2}^{infinity} 1/(k^(1/3)*log(k))。判断收敛/发散，并选能推出结论的 test。

  1. Use the memory rule for 1/(k^p log k).用 1/(k^p log k) 记忆规则。
  2. Here p=1/3<1, so log is too weak to save it.这里 p=1/3<1，log 太弱，救不了。
  3. Useful tests: comparison with p-series, or integral test.可用：和 p-series 比较，或者 integral test。
Conclusion: diverges
Tests: comparison test with p-series; integral test
不要选 ratio test：这种题 ratio 通常给 L=1，没结论。不要选 kth-term test：通项趋于 0。Do not choose ratio test: it usually gives L=1. Do not choose kth-term test: the term tends to 0.

Detailed Linear Algebra Walkthroughs 下面开始细讲：线代题型

Q2 Real Type Notes

Linear Transformation in R3真实题 2：线性变换拆向量

Real shape: T:R^3 -> R^2, T(e1)=<2,9>, T(0,1,1)=<4,-4>, later T(e3)=<-3,-7>.

真实题型：T:R^3 -> R^2，已知 T(e1)=<2,9>，T(0,1,1)=<4,-4>，后面又给 T(e3)=<-3,-7>。

  1. T(0,0,0)=<0,0> for every linear transformation.线性变换必有 T(0,0,0)=<0,0>。
  2. Before T(e3) is given, T(0,1,0) is undetermined.还没给 T(e3) 之前，T(0,1,0) 算不出，填 undetermined。
  3. (1,1,1)=e1+(0,1,1), so T(1,1,1)=<2,9>+<4,-4>=<6,5>.(1,1,1)=e1+(0,1,1)，所以 T(1,1,1)=<2,9>+<4,-4>=<6,5>。
  4. After T(e3) is given, T(e2)=T(0,1,1)-T(e3)=<7,3>.后面给了 T(e3) 后，T(e2)=T(0,1,1)-T(e3)=<7,3>。
(a) <0,0>
(b) undetermined
(c) <6,5>
after T(e3): T(0,1,0)=<7,3>
记忆：e1 管第一个数，e2 管第二个数，e3 管第三个数。Memory: e1 controls the first coordinate, e2 the second, e3 the third.
Q3 Real Type Free

Transformation Matrix From Graphs真实题 3：从图读 T(A), T(B)，再求矩阵 P

Real shape: left graph has vectors such as A=(-2,-2), B=(2,0); right graph gives T(A), T(B). Find alpha,beta and the matrix P.

真实题型：左图读原点坐标，比如 A=(-2,-2)，B=(2,0)；右图读 T(A)、T(B)。先找组合系数，再求矩阵 P。

  1. Express e1=(1,0) and e2=(0,1) using A and B.先把 e1=(1,0) 和 e2=(0,1) 用 A,B 拼出来。
  2. e1=0*A+(1/2)*B, so alpha1=0, beta1=1/2.e1=0*A+(1/2)*B，所以 alpha1=0，beta1=1/2。
  3. e2=(-1/2)*A+(-1/2)*B, so alpha2=-1/2, beta2=-1/2.e2=(-1/2)*A+(-1/2)*B，所以 alpha2=-1/2，beta2=-1/2。
  4. The columns of P are T(e1) and T(e2).P 的两列就是 T(e1) 和 T(e2)。
alpha1=0, beta1=1/2, alpha2=-1/2, beta2=-1/2
Maple matrix example: <<0,-1>|<-1,0>>
读图口诀：左图读 A,B 原坐标；右图读 T(A),T(B) 新坐标；矩阵列是 T(e1),T(e2)。Graph rule: left graph gives original vectors; right graph gives images; matrix columns are T(e1), T(e2).
Q4 Real Type Notes

Not Diagonalisable + Split Into A+B真实题 4：证明不可对角化 + 拆成两个可对角化矩阵

Real shape: Maple gives det(C-lambda*I)=(lambda-12)^2, R1=C-12*I, and Rank(R1)=1.

真实题型：Maple 给 det(C-lambda*I)=(lambda-12)^2，R1=C-12*I，且 Rank(R1)=1。

  1. The only eigenvalue is 12 with algebraic multiplicity 2.唯一特征值是 12，代数重数是 2。
  2. For a 2x2 matrix, nullity(R1)=2-rank(R1)=1.2x2 中，nullity(R1)=2-rank(R1)=1。
  3. Only one independent eigenvector, so not diagonalisable.只有 1 个独立特征向量，不够 2 个，所以不可对角化。
  4. For A+B=C, pick diagonal A, then let B=C-A and check B has two distinct eigenvalues.拆 A+B=C：先选一个 diagonal 的 A，再令 B=C-A，检查 B 有两个不同特征值。
Essay phrase: algebraic multiplicity 2 but eigenspace dimension 1, so C is not diagonalisable.
[A,B] example: [<<1,0>|<0,0>>, <<9,4>|<-1,14>>]
记忆：重复特征值不可怕，怕的是独立特征向量不够。Memory: repeated eigenvalue is not the problem; not enough independent eigenvectors is the problem.
Q5 Real Type Free

Span, Not Span, Independent, Dependent, Basis真实题 5：span / not span / independent / dependent / basis 定义题

Real shape: choose dropdown phrases for sets in R^2 and R^5.

真实题型：给你 R^2 和 R^5 里的几组向量，让你选下拉框证明 span、不 span、线性无关、线性相关，并判断 basis。

  1. Span: for every vector x, there exist coefficients.span：对每个目标向量 x，都存在一些系数能拼出来。
  2. Not span: there exists a non-zero vector y that cannot be made by any coefficients.not span：存在一个非零 y，不管怎么选系数都拼不出来。
  3. Independent: combination equals 0 only when all coefficients are 0.线性无关：凑 0 只能全 0。
  4. Dependent: there is a non-zero coefficient choice that makes 0.线性相关：存在不全为 0 的系数，也能凑成 0。
  5. Basis for R^n needs exactly n vectors, spanning, and independent.R^n 的 basis 必须刚好 n 个，还要 span 和 independent。
Span: for every x, for some coefficients
Not span: there exists non-zero y, for no choices of coefficients
Independent: v=0 only all coefficients 0
Dependent: w=0 for some coefficients not all zero
记忆：R 几就要几个 basis vectors。R^5 只有 3 个向量，一定不够。Memory: R^n needs n basis vectors. Three vectors can never be a basis for R^5.
Q6 Real Type Notes

Polynomial Vectors and Pivot Columns真实题 6：多项式看成向量，pivot columns 找 basis

Real shape: polynomials in P2, such as p1=-20-80x-160x^2, p2=9+36x+72x^2, plus e1=1, e2=x, e3=x^2. Row reduction gives pivot columns.

真实题型：P2 里的多项式，比如 p1=-20-80x-160x^2，p2=9+36x+72x^2，再加 e1=1、e2=x、e3=x^2。题目给 row reduction，让你找 basis。

  1. Convert polynomial a+b*x+c*x^2 to coefficient vector <a,b,c>.把多项式 a+b*x+c*x^2 看成系数向量 <a,b,c>。
  2. In echelon form, find the first non-zero entry in each row.在阶梯矩阵中，找每一行第一个非零数。
  3. Those columns are pivot columns. If pivots are columns 1,3,4, the answer is {p1,p3,p4}.这些列就是 pivot columns。如果 pivot 是第 1、3、4 列，答案就是 {p1,p3,p4}。
dimension of span = number of pivots = 3
basis subset = {p1,p3,p4}
记忆：pivot column 一定回去找原矩阵/原列表里的列名，不是找化简后那列。Memory: pivot columns refer back to the original column names.
Question Shape题型长相 What To Remember背什么 First Line To Write第一步写什么
Linear transformation with e1,e2,e3线性变换给 e1,e2,e3 T(a+b)=T(a)+T(b); T(0)=0.T 可以拆加法；T(0)=0。 (0,1,1)=e2+e3
Matrix of a transformation from graph从图读变换矩阵 Columns of P are T(e1), T(e2).矩阵 P 的两列就是 T(e1), T(e2)。 e1 = alpha*A + beta*B
Span / not span张成 / 不张成 Span: every target can be made. Not span: one target cannot be made.张成：每个目标都能拼。不张成：存在一个目标拼不出。 for every x, there exist coefficients
Independent / dependent线性无关 / 线性相关 Independent: only all-zero coefficients make 0. Dependent: non-zero coefficients can make 0.无关：凑 0 只能全 0。相关：不全 0 也能凑 0。 c1*v1+c2*v2+...=0
Basis in R^nR^n 的 basis A basis has exactly n vectors, spans, and is independent.basis 必须刚好 n 个，还要 span 和 independent。 R^5 needs 5 vectors
Polynomial vectors and pivots多项式向量和 pivot Use coefficient vectors. Pivot columns give basis columns.把多项式看成系数向量；pivot columns 对应 basis。 p=a+b*x+c*x^2 -> <a,b,c>
Diagonalisation A=MDM^-1对角化 A=MDM^-1 Column i of M must match diagonal entry i of D.D 第几个特征值，M 第几列就必须放对应特征向量。 D_ii lambda -> M column i eigenvector
Not diagonalisable不可对角化 Repeated eigenvalue needs enough independent eigenvectors.重复特征值时，要检查独立特征向量够不够。 nullity(C-lambda*I)

Detailed Q9 and Q12-Q15 Walkthroughs 下面开始细讲：Q9 和 Q12-Q15 补充题型

Q9 Real Type Free

Diagonalisation Options: Match D and M真实题 9：对角化选项题，D 和 M 必须一一配对

真实题型：给出 lambda=9 的两个特征向量方向，和 lambda=-4 的一个特征向量方向，让你从多个 A=M D M^(-1) 选项中选正确的。

  1. 看 D 的第 1 个对角线值，M 第 1 列必须是它对应的 eigenvector。Look at the first diagonal entry of D; first column of M must be its eigenvector.
  2. 第 2 个、第 3 个同理，顺序不能乱。Repeat for the second and third entries. Order matters.
  3. 还要检查 M 的列不能同方向。比如第二列是第一列 2 倍，则 M 不可逆。Also check columns are independent. If column 2 is twice column 1, M is not invertible.
Rule 1: D_ii = lambda means M column i must be a lambda-eigenvector.
Rule 2: reject if M columns are linearly dependent.
秒题读法：如果 D=diag(9,9,-4)，那么 M 的前两列必须来自 lambda=9 的特征向量，第三列必须来自 lambda=-4 的特征向量。 如果 M 里两列同方向，比如 (-2,-2,6)=2(-1,-1,3)，直接错，因为 M 不可逆。 Fast read: if D=diag(9,9,-4), the first two columns of M must be 9-eigenvectors and the third column must be a -4-eigenvector. If two columns point in the same direction, for example (-2,-2,6)=2(-1,-1,3), reject because M is not invertible.
Q12 Real Type Notes

Graph Cross-Sections: Identify Powers and Signs真实题 12：从截面图判断指数和符号

真实题型：表达式类似 p*x^a - y + r*z^b = E，给你不同截面图，判断 a,b,p,r,E 的符号或大小。

  1. 看 z=0 的 x-y 图：变成 y=p*x^a-E。At z=0, the equation becomes y=p*x^a-E.
  2. 图是直线，指数是 1；图是抛物线，指数是 2。Line means power 1; parabola means power 2.
  3. 开口向上说明对应系数正；开口向下或相反方向说明系数负。Opening upward means positive coefficient; opposite opening means negative.
  4. 最低点如果是 y=-2，且 y=p*x^2-E，则 E=2。If the vertex is y=-2 in y=p*x^2-E, then E=2.
Typical result from this seen type: a=2, b=2, p>0, r<0, |p|>1, E=2
Q13 Test Template Free

Convergence Test Selection: Fast Recognition真实题 13：级数判别法快速识别

Real shape: choose which convergence test applies from the form of the series.

真实题型：不一定要长算，先从级数长相判断应该用哪个 test。

ShapeUseMemory
(-1)^k a_kAlternating / Leibniz正负交替、越来越小、极限 0
n! or powers like 3^nRatio testL<1 收敛，L>1 发散，L=1 没结论
1/n^pp-seriesp>1 收敛，p<=1 发散
记忆：先看有没有 (-1)^n，再看有没有 ! 或指数，最后看是不是 1/n^p。Memory: check alternating sign first, then factorial/exponential, then p-series.
Q14 Test Template Notes

Convergence Test Selection: Log and Borderline Cases真实题 14：log 和边界级数判别

Real shape: decide convergence for series involving log(k), comparison, integral test, or a ratio-test trap.

真实题型：遇到 log(k)、comparison、integral test，或者 ratio test 给 L=1 的陷阱。

ShapeUseMemory
1/(k^p log k)comparison / integral先看 p；p<1 发散，p>1 收敛；p=1 再看 log 幂
log(k) < k^acomparisonlog 长得比任何正幂慢，可用来和 p-series 比
ratio gives L=1not usefulL=1 不能推出收敛或发散
term limit not zerok-th term test通项不趋 0 才能直接发散
For 1/(k^(1/3)log(k)): diverges; usable tests are comparison with p-series or integral test.
Q15 Core Formula Free

Eigenvalue and Eigenvector Meaning真实题 15：特征值/特征向量和对角化概念

核心公式是 A*v=lambda*v。矩阵作用后方向没变，只是被拉长/缩短了 lambda 倍。

  1. 求 eigenvalue：解 det(A-lambda*I)=0。Find eigenvalues by solving det(A-lambda*I)=0.
  2. 求 eigenvector：把 lambda 代回 (A-lambda*I)v=0。Find eigenvectors by solving (A-lambda*I)v=0.
  3. 对角化：必须有足够多的独立 eigenvectors。Diagonalisation needs enough independent eigenvectors.
  4. 选 A=MDM^(-1)：D 第几个特征值，M 第几列就放对应特征向量；并且 M 的列不能同方向。For A=MDM^(-1), diagonal entry i in D must match column i in M, and columns of M cannot repeat directions.
Av = lambda*v
eigenvector = direction unchanged
eigenvalue = scale factor
A=MDM^(-1): D order must match M column order

Detailed Q16-Q25 Walkthroughs 下面开始细讲：Q16-Q25 新增核心题型

Q16 Real Type Notes

Conditional Probability With Routes真实题 16：路线概率 + given 后除以条件世界

真实题型：selector 骰子先选红或蓝，再看对应颜色骰子是否 WIN。问赢的概率、已知赢了 selector 是红的概率、已知至少一个 coloured die 赢了但玩家没赢的概率。

Real shape: a selector die chooses red or blue, then the matching coloured die decides WIN. Compute total win probability and conditional probabilities.

  1. 概率先读成“想要的面数/总面数”：P(red selector)=1/6, P(blue selector)=5/6, P(red WIN)=3/20, P(blue WIN)=1/4。Read probabilities as wanted faces divided by total faces.
  2. 赢 = 红路线 + 蓝路线：1/6*3/20 + 5/6*1/4 = 7/30。Winning equals red route plus blue route.
  3. given won 问红：红赢路线 / 全部赢 = (1/40)/(7/30)=3/28。Given won, divide the red-win route by total win probability.
  4. 至少一个 = 1-一个都没有。没赢但至少一个赢 = selector 选错中奖颜色。At least one means one minus none; not win means the selector chose the wrong winning colour.
P(win)=7/30
P(selector red | win)=3/28
P(not win | at least one coloured WIN)=31/87
背法：路线内部用乘法；路线之间用加法；看到 given 最后一定除以 given 的概率。Memory: multiply inside a route, add separate routes, and divide by the given event.
Q17 Real Type Free

Normal Distribution: Probability and Inverse Probability真实题 17：正态分布标准化和反查 z

真实题型：茶包重量近似 normal，mean 是 mu，standard deviation 是 4。先问 mu=566 时低于 560 的比例，再问低于 560 的比例最多 27/10000 时该设什么 mu。

  1. 题目写 mean mu, standard deviation 4，做题只抓：mean=mu, sd=4。Use mean=mu and sd=4.
  2. 问概率：Z=(560-566)/4=-1.5，所以 P(X<560)=P(Z<-1.5)=0.0668。For probability, standardise and read the left area.
  3. 问 mean：先把 27/10000=0.0027 反查成 z=-2.78215。For mean, use inverse normal to find the z-value for 0.0027.
  4. 解 (560-mu)/4=-2.78215，得到 mu=571.1286。Solve the standardisation equation for mu.
P(X<560)=0.0668
mu=571.1286
背法：给 x 求概率用 CDF；给概率求 x 或 z 用 Quantile/invNorm。连续正态题不用 continuity correction。Memory: x to probability uses CDF; probability to z uses Quantile/invNorm. No continuity correction for a continuous variable.
Q18 Real Type Notes

Homogeneous ODE: y=xv Then Separate真实题 18：齐次型微分方程，令 y=xv

真实题型：x^3 dy/dx = y^3+2x^2y。先判断类型，再用 y=xv(x) 变成可分离方程。

  1. 最高只有 dy/dx，所以是 first order；有 y^3，所以不是 linear。Only dy/dx appears, so it is first order. The y^3 term makes it non-linear.
  2. 变成 dv/dx=(v^3+v)/x 后，可以写成 1/(v^3+v) dv = 1/x dx，所以 DE2 是 separable。After substitution, it separates.
  3. 部分分式必背：1/(v^3+v)=1/v - v/(v^2+1)。Memorise the partial fraction split.
  4. 积分后：log|v| - 1/2*log(v^2+1)=log|x|+D，所以 h(v)=-1/2*log(v^2+1)。Integrate and match the given form.
Original DE: first order only
DE2: first order and separable
h(v) = -1/2*log(v^2+1)
背法：看到 dy/dx 最高一次叫 first order；看到 y^3 不叫 linear；能把 v 和 x 分两边才叫 separable。Memory: first order is about derivative order, not powers of y.
Q19 Real Type Free

Exponential Substitution and Partial Fractions真实题 19：指数换元 + 部分分式

真实题型：integral 1/(e^(10x)-2e^(5x)) dx，令 u=e^(5x)。

  1. 秒法：u=e^(5x) 时，dx=du/(5u)，所以新分母 = 5u 乘原分母换成 u。With u=e^(5x), dx=du/(5u), so multiply the substituted denominator by 5u.
  2. 原分母 e^(10x)-2e^(5x) 变 u^2-2u，所以 f(u)=5u(u^2-2u)=5u^2(u-2)。The denominator becomes 5u^2(u-2).
  3. u^2 给两项：A/u+a/u^2；u-2 给一项：C/(u-2)，一共 3 项。Repeated u^2 gives two partial-fraction terms.
  4. 找 a/u^2 的 a：乘 u^2 后代 u=0，a=1/[5(0-2)]=-1/10。To get the u^-2 coefficient, multiply by u^2 and set u=0.
f(u)=5*u^2*(u-2)
number of terms = 3
a = -1/10
A = 20, B = 2
背法：u=e^(5x) 就多乘一个 5u 到分母。Memory: u=e^(5x) adds a 5u factor in the denominator.
Q20 Real Type Notes

First-Order Linear ODE: Integrating Factor真实题 20：一阶线性 ODE 的 integrating factor

真实题型：sqrt(x^2+25)y' + x^3y = x^2+25。问 integrating factor 里的 g(x)、推荐换元、以及 h(0)。

  1. 固定第一步：先除掉 y' 前面的东西，变成 y'+g(x)y=r(x)。First divide by the coefficient of y'.
  2. 所以 g(x)=x^3/sqrt(x^2+25)。The coefficient of y is g(x).
  3. 看到 sqrt(x^2+a^2)，推荐 x=a*tan(theta) 或 x=a*sinh(theta)。这里是 5。For sqrt(x^2+a^2), use tan or sinh substitutions.
  4. h=exp(integral g)；本题 h(0)=exp(-250/3)。The integrating factor is exp(integral g).
g(x)=x^3/sqrt(x^2+25)
choose x=5*tan(theta) and x=5*sinh(theta)
h(0)=exp(-250/3)
背法：线性 ODE 一定先整理成 y'+g(x)y=...，不要直接把原式里的 x^3 当 g。Memory: g is after y' has coefficient 1.
Q21 Real Type Free

Surface Area Bounds From Graphs真实题 21：旋转曲面面积上下界估算

真实题型：给 3<=f(x)<=8 和 0.31<=f'(x)<=0.59，用 A=integral 2*pi*f(x)*sqrt(1+(f')^2) dx 做上下界。

  1. (a) 粗估：f 用常数 3 和 8，f' 用 0.31 和 0.59。常数积分就是乘区间长度 10。For rough bounds, replace f and f' by their constant lower/upper bounds.
  2. (a) 得 197 <= A <= 584。The rough bound is 197 to 584 after rounding outwards.
  3. (b) 更准：f(x) 用线性界 3+0.31x 和 3+0.59x，sqrt 那坨还是用常数界。For better bounds, use linear bounds for f(x) and constant bounds for the square-root factor.
  4. 必背积分：integral_0^10 (3+kx) dx = 30+50k。Memorise integral_0^10 (3+kx) dx = 30+50k.
(a) 197 <= A <= 584
(b) 299 <= A <= 435
背法：a 和 b 都是在积分；a 积分常数，b 积分一次函数。Memory: both parts integrate; part (a) integrates constants, part (b) integrates a linear function.
Q22 Real Type Notes

Taylor Polynomials: Read Signs and Error Bounds真实题 22：Taylor 图像读导数正负 + 误差界

真实题型：给 x0=0.3 附近的 p0 到 p5 图像，判断各阶导数正负，再用 Lagrange remainder 估 p5(0.5) 的误差。

  1. 系数口诀：a_n=f^(n)(x0)/n!，所以系数正负就是对应导数正负。Coefficient sign matches derivative sign because a_n=f^(n)(x0)/n!.
  2. 图像口诀：偶数看两边，奇数看右边。上正，下负，水平是 0。Even powers: look at both ends. Odd powers: look to the right. Up is positive, down is negative.
  3. 本题：f'=0，f''<0，f'''<0，f^(4)>0，f^(5)>0。This gives zero, negative, negative, positive, positive.
  4. p5 误差看下一阶 f^(6)。若最大在 x0，直接用 |a6|*|0.5-0.3|^6。A p5 error uses the sixth derivative. If the maximum is at x0, use |a6||x-x0|^6.
zero: f'(0.3)
positive: f^(4)(0.3), f^(5)(0.3)
negative: f''(0.3), f'''(0.3)
error bound = 6.8*(0.2)^6 = 0.0004352
f is a polynomial of degree 7
背法：x0=0.3 来自 “Taylor around x0=0.3”；要估 f(0.5)，距离就是 0.5-0.3。Memory: the distance in the error formula is from the Taylor centre to the target x.
Q23 Real Type Free

Cross-Sections, Tangent Plane, and Linear Approximation真实题 23：截面图、切平面和线性近似

真实题型：左图是 y=b 截面，右图是 x=a 截面。左图斜率 -3.4，右图斜率 2.1。

  1. LEFT 固定 y=b，所以看 x 方向：Fx=-3.4，曲线是 z=F(x,b)。Left fixes y=b, so its slope is Fx and the section is z=F(x,b).
  2. RIGHT 固定 x=a，所以看 y 方向：Fy=2.1，曲线是 z=F(a,y)。Right fixes x=a, so its slope is Fy and the section is z=F(a,y).
  3. 方向向量口诀：LEFT <1,0,slope>，RIGHT <0,1,slope>。Direction vector mantra: left is <1,0,slope>, right is <0,1,slope>.
  4. 切平面必背：z=z0+Fx(x-x0)+Fy(y-y0)。法向量可用 <-Fx,-Fy,1>。Tangent plane formula: z=z0+Fx(x-x0)+Fy(y-y0). A normal is <-Fx,-Fy,1>.
  5. 线性近似：Delta F≈Fx Delta x+Fy Delta y。Linear approximation uses Delta F approximately Fx Delta x plus Fy Delta y.
Fx=-3.4, Fy=2.1
LEFT parallel vector: <10,0,-34>
RIGHT perpendicular vector: <0,21,-10>
normal c: <34,-21,10>
Delta F ≈ 2.36
背法：LEFT 固定 y 看 Fx；RIGHT 固定 x 看 Fy。切平面公式一定背 z=z0+Hx(x-x0)+Hy(y-y0) 这个形状。Memory: left gives Fx, right gives Fy.
Q24 Real Type Notes

Standard Normal Table With Conditional Probability真实题 24：标准正态表 + 条件概率

真实题型：求 P(Z<2.43 | Z>1.43)，表给的是 P(Z<=z) 左侧面积。

  1. 条件概率先改写：P(Z<2.43 | Z>1.43)=P(1.43<Z<2.43)/P(Z>1.43)。Condition on the right tail, then divide by the right-tail probability.
  2. 查 2.43：行 2.4，列 0.03，得 0.9925。For 2.43, use row 2.4 and column 0.03.
  3. 查 1.43：行 1.4，列 0.03，得 0.9236。For 1.43, use row 1.4 and column 0.03.
  4. 答案：(0.9925-0.9236)/(1-0.9236)=0.9018。Middle area divided by right tail.
(Phi(2.43)-Phi(1.43))/(1-Phi(1.43)) = 0.9018
背法：表里永远是左面积；右尾巴 = 1-左面积；中间面积 = 大左面积 - 小左面积。Memory: table is left area; right tail is 1-left area; middle is big-left minus small-left.
Q25 Real Type Free

Exact ODE, Level Curves, and Tangent Planes真实题 25：exact ODE、level curve 和切平面

真实题型：16x^3y^4 + (16x^4y^3-8sin(4y))dy/dx=0，并给 H(x,y)=2 的 level curve。

  1. 读成 M+N*y'=0，其中 M=Hx=16x^3y^4，N=Hy=16x^4y^3-8sin(4y)。Read M+N y'=0 as M=Hx and N=Hy.
  2. 检查 exact：M_y=N_x=64x^3y^3，所以选 first order 和 exact。Check exact by comparing M_y and N_x.
  3. 曲面 z=H(x,y) 的法向量：<Hx,Hy,-1>。For z=H(x,y), a normal is <Hx,Hy,-1>.
  4. 切平面必背：z=z0+Hx(x-x0)+Hy(y-y0)。Use z=z0+Hx(x-x0)+Hy(y-y0).
  5. 找 H：先对 Hx 按 x 积分，H=4x^4y^4+C(y)；再用 Hy 对出 C(y)=2cos(4y)+K。Recover H by integrating Hx with respect to x, then matching Hy.
ODE type: first order, exact
normal at (4,3,2): <82944,110592-8*sin(12),-1>
tangent plane at (3,4,-5): z=-5+110592*(x-3)+(82944-8*sin(16))*(y-4)
H=4*x^4*y^4+C(y), C(y)=2*cos(4*y)+K
背法：z=H(x,y) 的切平面就用 z=z0+Hx(x-x0)+Hy(y-y0)；exact ODE 的 M,N 就是 Hx,Hy。Memory: exact ODE gives the partial derivatives of H.

Memory Sheet 最后背诵表

  • Converges means the infinite process has a finite answer.converges = 最后有有限答案。
  • Ratio test: L<1 converges, L>1 diverges, L=1 no conclusion.ratio test：L<1 收敛，L>1 发散，L=1 没结论。
  • Alternating test: alternating, decreasing, limit 0.alternating test：正负交替、越来越小、极限为 0。
  • Continuous probability: intervals matter, single points have probability 0.连续概率：区间才有面积，单点概率为 0。
  • Normal table: table values are left areas. Conditional normal means middle area divided by the given area.正态表：表里是左面积。条件正态 = 中间面积 / given 的面积。
  • Surface area bounds: replace f and f' by lower or upper bounds, then integrate.曲面面积上下界：把 f 和 f' 换成上下界，再积分。
  • Taylor signs: even powers look at both ends; odd powers look at the right end.Taylor 图像读符号：偶数看两边，奇数看右边。
  • Tangent plane: z=z0+Hx(x-x0)+Hy(y-y0).切平面：z=z0+Hx(x-x0)+Hy(y-y0)。
  • Linear approximation: Delta F is approximately Fx Delta x plus Fy Delta y.线性近似：ΔF≈FxΔx+FyΔy。
  • Exact ODE: M+N y'=0 gives M=Hx and N=Hy.exact ODE：M+N*y'=0 读成 M=Hx, N=Hy。
  • Diagonalisation: D order must match M column order; M columns cannot repeat directions.对角化：D 的顺序必须对应 M 的列顺序；M 的列不能重复方向。
  • Maple syntax: use * for multiplication and <a,b> for vectors.Maple 写法：乘法必须写 *，向量用 <a,b>。

Practice These Templates用 40 题检测这些模板

The quiz focuses on recognising which method to use, not long computation.

测验重点是“看到题先选方法”，不是长计算。

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